Uniqueness theorem for inverse scattering problem with non-overdetermined data

TL;DR

This paper proves that the scattering data for all directions and positive energies uniquely determine a compactly supported real-valued potential in three-dimensional inverse scattering problems.

Contribution

It establishes a uniqueness theorem for inverse scattering with non-overdetermined data, extending the understanding of data sufficiency for potential recovery.

Findings

Scattering data for all directions and energies determine the potential uniquely.

The proof applies to sufficiently smooth, compactly supported potentials.

The result reduces the amount of data needed for unique reconstruction.

Abstract

Let $q(x)$ be real-valued compactly supported sufficiently smooth function, $q\in H^\ell_0(B_a)$, $B_a:=\{x: |x|\leq a, x\in R^3$ . It is proved that the scattering data $A(-\beta,\beta,k)$ $\forall \beta\in S^2$, $\forall k>0$ determine $q$ uniquely. here $A(\beta,\alpha,k)$ is the scattering amplitude, corresponding to the potential $q$.